Research study-the golden ratio of life

The extensive application of the golden ratio-research background and purpose: Mathematics is a magical subject, and its extensive application has become an indispensable part of life. Whether it is building engineering, calculating probability or statistics, the shadow of mathematics can be seen everywhere. This time, we chose the golden section ratio in geometry as an example to understand the wide application of the golden section ratio in real life, and to understand the significance of the golden section from different forms of works of art, photos and excellent buildings, and to experience the charm of mathematics and the magic of its application. Second, research methods and process: 1, data survey: The three of us worked together and consulted Mr. Yi Nanxuan's Beauty of Mathematics, as well as Mr. Meng Qingfei's Golden Section, Mathematics Newspaper, Mathematical Science and other books. 2. Measurement experiment: We measured and analyzed some real examples in life with our own understanding of the golden ratio. For example, we measure golden rectangles such as doors, windows and desktops to see if they meet the golden ratio. The following are some data: the length (meter) of the window calendar TV is 0.98 1 0.60.780.8, the width (meter) is 0.610.480.5, and the ratio between them is 0.6120.625 0.6150.60. The approximate value of 1.6 18 is 0.6 18, which is actually the so-called golden section number. The ratio of 0.6 18 and 1854 was formally defined as "golden section law" by German and American scientist Cai Xin. The word "gold" means that people cherish it. The medieval mathematician Kepler called the golden section law and Pythagorean theorem "two treasures in geometry". Pachouri, a Venetian mathematician in the19th century, praised the golden section law as "the proportion given by God". 2) essence: divide a unit length line segment into two segments, so that "all segments/large segments = large segments/small segments" is as shown in the figure: then we say AC/BC=BC/AB. B is the golden section of line segment AC, and this ratio is called golden section ratio. If the length of BC is assumed to be x, and the length of AB is 1-x, then 1/x=x/ 1-x, x=- 1 3) can be used to draw the golden ratio: as shown in the figure, a right triangle is needed, and the two right sides AC and BC are 650 respectively. If AE=AD is taken on AC (it is known that the radius of the same circle is equal), then point E is the golden section of line segment AC. 2. The application of the golden ratio: Through observation, we find that the golden ratio is almost everywhere in life. Whether it is reflected in architecture, art, human body, photography, musical instruments, or the wide application of the golden section as a long and wide golden giant rectangle, it fully embodies the charm of the golden section. 1) The "golden ratio" in architecture can be traced back to the largest Egyptian pyramid of khufu built 4,600 years ago. The tower height146m, the square side at the bottom is 232m, and the ratio of 0.629 is about 5: 8. Before 2004, the Parthenon dedicated to Athena was built by the ancient Greeks on the Acropolis Mountain in south-central Athens. The aspect ratio of the front is gold. 1976 The Toronto TV Tower in Canada is 553.3 meters high, while its seven-story working hall is built in the air of 340 meters, and the ratio of 340: 553 is about 0.6 15. Not only in ancient times, but also in modern architecture, the golden ratio has always been regarded as an indispensable and important factor in architectural design, which makes the building look more harmonious and pleasing to the eye. 2) The Golden Section of the Human Body Italian mathematician Phippis first noticed the "unpopular" in mathematics-the golden section of the human body: he said that the length ratio of the upper and lower navel of the average human body is 0.6 18: 1, or similar to it. This is the best number of upper and western structures in the human body. In addition, he also found that there are three golden sections in the human body structure. The upper limb is divided into elbow joint, knee below navel and throat above navel. If the structural proportion of each part of a person conforms to the golden section law, it is the most standard body shape. For example, the image of Apollo, the sun god in ancient Greek mythology, and the statue of Venus, the goddess, respectively represent the typical physical beauty of men and women, which completely conforms to the golden section law and is absolutely wonderful. 3) Photography: Where should the main part of the photo be arranged to look good? The most commonly used method in photography is the golden section method, that is, the center of interest of the photo is determined at the position of 0.6 18 in the whole picture. 4) Art: The Mona Lisa is one of Leonardo da Vinci's most famous works. In this painting, Leonardo da Vinci applied the golden ratio of human body structure to figure painting, and achieved excellent artistic effect, making it a masterpiece handed down from generation to generation; Similarly, Leonardo da Vinci's famous painting "The Vitruvian Man" fully meets the requirements of the golden ratio and brings a wonderful sense of coordination to the audience, so it has become another world art classic. 5) Musical instrument: PInthagoras, an ancient Greek mathematician and philosopher, passed by a blacksmith's shop one day and was attracted by the crisp and pleasant sound of striking the iron. He stopped to listen carefully and intuitively concluded that the voice was "secret"! He walked into the workshop and carefully measured the dimensions of the anvil and the hammer, and found that the ratio between them was close to 1: 0.6 18. This discovery is still the scientific basis for making all kinds of musical instruments. 6) Application of golden graphics: ① Golden rectangle: The golden rectangle refers to the rectangle with the aspect ratio of golden number. In modern times, the golden rectangle has penetrated into every family. For example, the desktop of a writing desk, calendars on the wall, envelopes, TV screens, books, doors and windows, and even matchboxes are all golden rectangles. This shows people's preference for the golden rectangle. ② Five-pointed star pattern; The national flag, national emblem and military flag of our country all adopt the five-pointed star pattern, and it is found that the emblem of Pythagoras school with golden rectangle is also the five-pointed star pattern. What is the beauty of the five-pointed star? Divide the circle into five equal parts and connect it with another point in turn to form a regular pentagram. The beauty of this five-pointed star lies in the golden ratio of five sides (F and G in the figure are AC golden ratio points). This is the most symmetrical proportion, which can give people beautiful motivation. That's why the five-pointed star has such great charm. 1) Summary: The role of the golden section is unparalleled. Artists found that following the golden section to design the human body image, the human body will present the most beautiful figure. Musicians find that putting fingers on the golden section of strings will make music louder and more harmonious. Architects found that following the golden section to design a palace will make it more majestic and solemn, and designing a villa will make people feel more comfortable. Scientists have found that applying the golden section to production practice and scientific experiments can achieve remarkable economic benefits. The application of golden section is extremely extensive, and it is worthy of being a great treasure of geometry. Student experience: Through research study, we got an opportunity to explore and feel science by ourselves, and suddenly became interested in mathematics learning. At the same time, we also realize the importance of mathematics in our life, which makes us eliminate the wrong idea that "mathematics is boring and complicated and has no practical significance". For the content of the golden ratio, we have a deeper understanding. We are surprised at its magical function and also find its charm. So this study has benefited us a lot. In the future, we should actively participate in more investigation and study, learn knowledge in practice, exercise the ability to solve problems and learn to learn more.